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Math Trek

Ancient Islamic Penrose Tiles

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When Peter J. Lu traveled to Uzbekistan, he had no idea of the mathematical journey that he was about to embark on as well.

The Harvard graduate student in physics was fascinated by the beautiful and intricate geometric "girih" patterns on the 800-year-old buildings there, and he wanted to know how ancient artisans had created them. He discovered more than just a clever construction method. He also found an entirely unexpected level of mathematical sophistication in the designs, pointing at mathematical ideas that weren't formally developed until hundreds of years later.

Lu's determination to find out took him on a journey through hundreds of photographs of Islamic architecture in the libraries at Harvard—and now it's landed him an article in Science.

The only mathematical tools the builders had available to them were straightedge and compass. Theoretically, all these patterns could be made by drawing the lines directly onto the buildings.

But Lu noticed that the patterns were astonishingly perfect, even over very large areas. If the builders had been scribing the patterns directly on a wall, Lu expected the patterns to accumulate small errors that would be detectable on really big walls.

But he didn't see any errors. So he figured that they must have had some tricks to guide the pattern making, and he decided to figure out what they were.

He had a clue where to look from his undergraduate research. The patterns on the Islamic buildings reminded him of Penrose tiles, which are two simple geometric shapes, usually a kite and a dart or a fat and a skinny rhombus (diamond). When laid down in a tiling, these pairs of tiles can cover a plane in a pattern that never repeats.

As a Penrose tiling spreads across a larger and larger surface, the ratio between the numbers of each type of tile approaches the golden ratio. The golden ratio (or mean) is the irrational number 1.618 . . . .

Penrose tilings also have fivefold rotational symmetry, the same kind of a symmetry that a five-pointed star has. If you rotate the whole pattern by 72 degrees, it looks just the same.

For his undergraduate thesis, Lu had looked for examples in the physical world of quasicrystals, materials that are thought to have crystal structures that are three-dimensional versions of a Penrose tiling. Physical quasicrystals have remarkable properties. For example, metal quasicrystals don't conduct heat very well, and a company is now developing a tough but slippery nonstick coating from quasicrystals.

The patterns on Islamic buildings had lots of pentagons and decagons and stars, geometric figures with fivefold symmetry. Lu immediately thought of Penrose tiles.

"I see a fivefold pattern and my eyes light up, and I try to decompose it into tiles," he says.

Lu returned to Harvard and studied photos, trying to deconstruct the patterns. He found a picture of a 15th-century architectural scroll from Istanbul, the Topkapi scroll, which was "like the AutoCAD manual for ancient times," Lu says.

The main, dark pattern of red and blue lines was very complex and nonrepeating. But underneath, he saw a fainter red pattern that broke the design up into five decorated tiles: a decagon, a pentagon, a hexagon, a bowtie, and a rhombus.

He had hit paydirt. It was just like a Penrose tiling.

When Lu looked at photographs of Islamic buildings, he found that he could break the patterns on their surfaces up into the same shapes, even though the shapes often weren't immediately visible. "I couldn't sleep for days," he said. "I skipped Christmas break to work on it."

Lu suggests that Islamic architects used these shapes, which he calls girih tiles, to scribe the patterns onto the walls. That would explain how they tiled large surfaces with such precision.

Lu also figured out that the girih tiles could be broken up into the kites and darts of Penrose tiles. When he divided the tiles in this way, one building, the Darb-i Imam shrine, had a near-perfect Penrose tiling. The shrine was built in 1453, and it would be another 500 years before the mathematics behind Penrose tiles was developed.

The Darb-i Imam shrine was particularly remarkable because it showed girih tile patterns at two different scales, so that large girih tiles were broken up into smaller girih tiles. In principle, by repeatedly scaling up the tiling in this way, they could have covered an arbitrarily large wall with a Penrose tiling.

Lu has a history of finding math wherever he looks. In 2006, he turned his attention to the fossil record, creating a mathematical model that demonstrated that Earth's biosphere recovered from mass extinctions more quickly than people had thought. He published the result in the Proceedings of the National Academy of Sciences.

And in 2004, he landed his first publication in Science when he noticed that the spiral patterns in a Chinese jade ring from 500 B.C.E. were perfect Archimedes spirals—and showed that ancient Chinese technology must have been far more advanced than previously thought in order to produce such a ring.

Now all he needs to do is to finish his dissertation.


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